Kamae, T., Krengel, U. and O'Brien, G.L. (1977) Stochastic inequalities on partially ordered spaces, Ann. Probab. 5, 899--912.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....denote by T the invariant algebra. Then we have E[A(0) E[B(0) j T] a.s. 1) Furthermore, if A is independent of B then the conditional law of B on the event fE[B(0) j T] E[A(0) g is equal to the law of A. To prove this, the two ingredients are a representation theorem such as Theorem 1 in [14] and Birkho s Ergodic Theorem. The symbols and stand for is distributed as and is independent of , respectively. We use the notations N = N n f0g, R = R n f0g, and x = max(x; 0) x 0. For u; v in R or , u v denotes u(n) v(n) for all n. 3 The model We introduce ....

T. Kamae, U. Krengel, and G.L. O'Brien. Stochastic inequalities on partially ordered spaces. Annals of Probability, 5:899-912, 1977.

....on the class of probability measures on S. For a careful discussion on the matter of antisymmetry in a rather general setting for infinite S, see [4] The following characterization of stochastic ordering was established by Strassen [12] and fully investigated by Kamae, Krengel, and O Brien [5]. Suppose that there exists a pair (X 1 ; X 2 ) of S valued random variables [defined on some probability space( Omega ; F ; P) satisfying the properties X 1 X 2 (1.3) Received AMS 2000 subject classifications. Primary 60E05; secondary 06A06, 60J10, 05C38. Key words and phrases. Realizable ....

....four subclasses Classes B, Y, W and Z that partition the class of connected posets S. At this juncture we provide the reader with an overview of the main results of this paper. In Section 4 we present the first case of our investigation, where S is in Class B. Kamae, Krengel, and O Brien [5] showed that if A is a linearly ordered set then monotonicity equivalence holds for (A; S) We generalize this result (in our finite setting) to the case of an acyclic poset A (Theorem 4.1) see Section 2.1 for STOCHASTIC AND REALIZABLE MONOTONICITY 5 the definition of an acyclic poset. Theorem ....

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Kamae, T., Krengel, U., and O'Brien, G. L. (1977). Stochastic inequalities on partially ordered spaces. Ann. Probab. 5 899--912.

....probability measure P on (X X;B(X) B(X) with marginals P 1 and P 2 such that P (R ) 1. In many cases this happens to be equivalent to P 1 being stochastically smaller that P 2 . That is R fdP 1 R fdP 2 for all increasing continuous functions f : X R. We denote this as P 1 P 2 (see [26][25] Since this results holds in particular for compact Hausdor spaces, in the light of Proposition 2.3 we can obtain equivalent characterisations for stably compact spaces. We start by recalling some de nitions and auxiliary results. De nition 7.2 [42, Chapter II, Section 1] Let (X; G; be ....

T. Kamae, U. Krengel, and G. L. O'Brien, Stochastic inequalities on partially ordered spaces, The Annals of Probability 5 (1977), no. 6, 899-912.

.... 0, then the inequality in (i) is strict. Proof. In order to prove (i) it suces to show that EjT 1 1;n T 2 1;n j EjT 1 2;n T 2 2;n j for any two jointly stationary and ergodic interarrival times, T 1 1 and T 2 1 . We use an argument similar to that in Kamae, Krengel and O Brien [8], Theorem 8. Let T 3 1;n = max(T 1 1;n ; T 2 1;n ) and queue i, i = 1; 2 and 3, be the queue subject to the input T i 1 . Thus, jT 1 1;n T 2 1;n j = 2T 3 1;n T 1 1;n T 2 1;n : 5) Clearly, fT i 1;n ; n 1g, i = 1; 2 and 3 are also jointly stationary and ergodic. Moreover, ET ....

T. Kamae, U. Krengel and G.L. O'Brien, \Stochastic inequalities on partially ordered spaces," Ann. Prob., Vol. 5, pp. 899-912, 1977.

....establish stochastic comparisons for the interdeparture times D(k, n) in (2. 5) We say that a random element X 1 is stochastically less than or equal to another random element X 2 , and write X 1 st X 2 , if Eh(X 1 ) Eh(X 2 ) for all nondecreasing bounded measurable real valued functions h; see Kamae, Krengel and O Brien (1977). We are interested in the case X i is an array of real valued random variables. As before, let = d denote equality in distribution. Theorem 5.1. Suppose that the service times V(k, n) are all mutually independent. a) If V(k, n) d V(1 , n) for all k 1 and n 1, then D(k 1 , n) ....

Kamae, T., U. Krengel and G. L. O'Brien (1977). Stochastic inequalities on partially ordered spaces. Ann. Probab. 5, 899-912.

.... = Gamma(t) It therefore suffices to show that t P( Gamma(t) 2 Delta jT (1) t) is stochastically non decreasing, and for this it suffices to fix m 2 N and show that if Gamma m (t) Gamma(t)j B(0;m) then t P( Gamma m (t) 2 Delta jT (1) t) is stochastically non decreasing (see Kamae, Krengel and O Brien (1977, Proposition 2) By taking limits, one reduces this in turn to proving (3:2) t P( Gamma m (t) 2 Delta jT (n) t) is stochastically non decreasing for all n m: It suffices to consider (3.2) with m = n since decreasing m only weakens the conclusion. But (3.2) with m = n is precisely the ....

....5:1 n for all n 2 Z ; w 0: Proof. Define a Markov kernel q on (0; 1) by q( 0; y]jw) ae ae( 0; y=w] if w w 0 p( 0; y]jw 0 ) if w w 0 . Lemma 5.4 implies that q( Delta jw) OE p( Delta jw) for all w 0. This, the stochastic monotonicity of p( Delta jw) and a standard coupling argument (Kamae et al. 1977, Theorem 2) show that for each w 0 we may construct Markov chains (W n ) and (X n ) with transition kernels p and q, respectively, on the same probability space such that X 0 = W 0 = w and W n X n for all n 0. We abuse notation slightly and let Pw denote the underlying probability. Define a ....

Kamae, T., Krengel, U. and O'Brien, G.L. (1977). Stochastic inequalities on partially ordered spaces. Ann. Probab. 5, 899-912.

....(1 s j ) j ( g 1 j ( j of j ( hence P j ( j ( is also monotone in the coordinates k of . Note the similarity of equation (17) to (2) and (3) Nonparametric and Parametric IRT, and the Future 18 It follows immediately from Lemma 2 of Holland and Rosenbaum (1986; see also Kamae, Krengel O Brien, 1977) that for any non decreasing summary g(X) of X = X 1 ; X J ) E[g(X) j ] is nondecreasing in each coordinate ik of ; this implies the SOM (Stochastic Ordering of the Manifest score X by the latent trait) property of Hemker et al. 1997) that P [X c j ] is non decreasing in each ....

Kamae, T. Krengel, U., O'Brien, G. L. (1977). Stochastic inequalities on partially ordered spaces. Annals of Probability, 5, 899--912.

....function h k : IR mk IR , E[h k ( 1) 2) k) E[h k ( p (1) p (2) p (k) 3. 5) where (l) 1 (l) m (l) and (l) 1 (l) m (l) Having concluded these three steps, we may appeal to a result on coupling due to Kamae et al. [19] (Propositions 1 and 2) and to Kolmogorov s consistency theorem to show from (3.5) 8 the existence of a common probability space( Omega ; F ; P ) and clock sequences p , 0 p r, where each p is statistically indistinguishable from , and smaller than the corresponding p on every ....

Kamae, T., Krengel, U., and O'Brien, G. L. Stochastic inequalities on partially ordered spaces. The Annals of Probability 5, 6 (1977), 899--912.

.... is the integers and j(a) 1 for each a 2 A while (b) 1 for each b 2 B, then we get Hall s Marriage Theorem [Hal35] If we let j and be any integral values, we get a theorem of Hoffman [Ho60] see [Re84] If we let G be the reals, we get the finite version of the theorem of Strassen [St65] [KamKO77], or Major [LoP86, p. 76] If G is Q[ p 5] we get a funny beast indeed. Note that if we got Theorem 3.1 such for bipartite orders, the whole result follows immediately. Proof: First, suppose that minorizes j. The proof is by induction: suppose that the Theorem holds for all posets with fewer ....

T. Kamae, U. Krengel & G. L. O'Brien, Stochastic inequalities on partially ordered spaces, Ann. Prob. 5:6 (1977) 899--912.

....be a nite or countable family of partially ordered Polish spaces. Consider the product space K = Q i2I K i equipped with the product topology, the associated Borel algebra, and the coordinatewise partial order. Then K is a partially ordered Polish space. For a proof of the next result, see [19] or [27] Theorem 4.1.2, for the rst part and [27] Theorem 2.2.4. for the second part. 31 Proposition 8.3. Let K = Q i2I K i be the product space associated with a nite or countable family of partially ordered Polish spaces. For U; V 2 M(K) U st V ( U 1 i st V 1 i ; i 2 I; ....

....and E a partially ordered Polish space. Let be a measurable and increasing function from a subset of K = Q i2I K i to E. Then U st V = U 1 st V 1 : Equivalently, if u; v 2 L(K) are such that u U and v V , then u st v implies (u) st (v) The next result is proved in [19] and generalizes a classical theorem of Strassen. Proposition 8.4. Let U; V 2 M(K) be such that U st V . There exists a probability triple( F ; P ) and r.v. u and v in L(K) such that: u U; v V; u v; P a.s. In the same spirit, the following representation theorem is due to Skorohod ....

T. Kamae, U. Krengel, and G.L. O'Brien. Stochastic inequalities on partially ordered spaces. Annals of Probability, 5:899-912, 1977.

....then it would be achieved through the T (j) b equations. There is another significant difference in our approach. The monotonicity results in the other papers [CY, SY, TW] were stated in terms of the stochastic ordering st , where X st Y if and only if Prob(X z) Prob(Y z) for all real z [KKO]. For example, Cheng and Yao proved monotonicity in (I) by considering random variables S (j) b m and S (j) b m such that (S (j) b m ) n j=1 st ( S (j) b m ) n j=1 . In other words, they stochastically compare the performance of Q and Q 1 by picking one sample point each ....

T. Kamae, U. Krengel and G. O'Brien, Stochastic inequalities on partially ordered spaces, Annals of Probability 5 (1977), 899--912.

....random vectors S(X) S 1 (X) S n (X) 0 and S(Y ) S 1 (Y ) S n (Y ) 0 , one may, more generally, ask for conditions under which S(X) D S(Y ) 1. 12) For random vectors U and V , U D V if E I OE(U) E I OE(V ) holds for all bounded increasing functionals OE; see, e.g. Kamae, Krengel, and O Brien (1977) and Section 3 for equivalent conditions and further details. A necessary and sufficient condition is given also for the stochastic domination (1.12) Theorem 4) Interestingly, the condition turns out to be exactly the same as the condition for (1.3) Although one would, in general, expect the ....

....is defined by U D V if E I OE(U) E I OE(V ) holds for all bounded functionals OE which are increasing w.r.t. the componentwise vector ordering. Alternatively, U D V is characterized by P I fU 2 Ag P I fV 2 Ag holding for all Borel sets A with increasing indicator 1 A ( Delta) see, e.g. Kamae et al. 1977). The following simple result will also be used repeatedly. A.2. Let U , V , and T be random vectors, U and V taking values in a common Euclidean space. If, for some determinations of the conditional distributions, P I fU 2 AjT = tg P I fV 2 AjT = tg (3.1) holds for all Borel sets A with ....

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KAMAE, T., KRENGEL, U., and O'BRIEN, G.L. (1977). Stochastic inequalities on partially ordered spaces. Ann. Probab. 5 899--912.

.... 1 Gamma Theta sjF Theta g Pf Theta sg: We can therefore couple the Theta 1 Gamma Theta ; 0, with a sequence ae ; 0, of i.i.d. copies of Theta so that Theta n n Gamma1 X =0 ae ; n 1: This coupling is fairly standard; for a more general coupling result see Kamae, Krengel and O Brien (1977). Together with (4.7) and the strong law of large numbers this proves lim sup n 1 oe(n; r; n lim sup n 1 1 n n Gamma1 X =0 ae = E Theta: Xi ON SOME GROWTH MODELS 27 To prove the lower bound in (1.9) we apply Lemma 4.1 with C = 0; L] d Gamma Theta f0g ; where M = ....

T. Kamae, U. Krengel and G. L. O'Brien (1977), Stochastic inequalities on partially ordered spaces, Ann. Probab. 5, 899-912.

....vectors S(X) S 1 (X) S n (X) 0 and S(Y ) S 1 (Y ) S n (Y ) 0 , one may, more generally, ask for conditions under which S(X) D S(Y ) 1. 12) For random vectors U and V , U D V if E I OE(U) E I OE(V ) holds for all bounded increasing functionals OE; see, e.g. Kamae, Krengel, and O Brien (1977) and Section 3 for equivalent conditions and further details. A necessary and sufficient condition is given also for the stochastic domination (1.12) Theorem 4) Interestingly, the condition turns out to be exactly the same as the condition for (1.3) Although one would, in general, expect the ....

....is defined by U D V if E I OE(U) E I OE(V ) holds for all bounded functionals OE which are increasing w.r.t. the componentwise vector ordering. Alternatively, U D V is characterized by P I fU 2 Ag P I fV 2 Ag holding for all Borel sets A with increasing indicator 1 A ( Delta) see, e.g. Kamae et al. 1977). The following simple result will also be used repeatedly. A.2. Let U , V , and T be random vectors, U and V taking values in a common Euclidean space. If, for some determinations of the conditional distributions, P I fU 2 AjT = tg P I fV 2 AjT = tg (3.1) holds for all Borel sets A with ....

[Article contains additional citation context not shown here]

KAMAE, T., KRENGEL, U., and O'BRIEN, G.L. (1977). Stochastic inequalities on partially ordered spaces. Ann. Probab. 5 899--912.

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Kamae, T., Krengel, U. and O'Brien, G.L. (1977) Stochastic inequalities on partially ordered spaces, Ann. Probab. 5, 899--912.

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Kamae T., Krengel U. & O'Brien G.L., Stochastic inequalities on partially ordered spaces, Ann. Prob. 5 (1977), 899--912. 30

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T. Kamae, U. Krengel and G.L. O'Brien, \Stochastic inequalities on partially ordered spaces," Ann. Prob., Vol. 5, pp. 899-912, 1977.

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Kamae, T., Krengel, U. and O'Brien, G. L. (1977). Stochastic inequalities on partially ordered spaces. Ann. Probability 5 899-912.

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Kamae, T., Krengel, U. and O'Brien, G.L. (1977). Stochastic inequalities on partially ordered spaces. Annals of Probability 5, 899-912.

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T. Kamae, U. Krengel, and G. L. O'Brien. Stochastic inequalities on partially ordered spaces. The Annals of Probability, 5(6):899-912, 1977.

No context found.

Kamae, T., Krengel, U. and O'Brien, G.L. (1977). Stochastic inequalities on partially ordered spaces. Ann. Probab. 5, 899-912.

No context found.

Kamae, T., Krengel, U. and O'Brien, G. L. (1977), Stochastic inequalities on partially ordered spaces, Annals of Probability 5, 899--912.

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T. Kamae, U. Krengel, and G. L. O'Brien. Stochastic inequalities on partially ordered spaces. The Annals of Probability, 5(6):899--912, 1977.

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